To find the domain of a function from its graph, you identify all the possible x-values (input values) that the graph covers.
Here's a breakdown of how to do it:
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Visualize the x-axis: The domain represents the set of all x-values for which the function is defined. Focus your attention on the horizontal axis (x-axis).
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Project the graph onto the x-axis: Imagine shining a light directly down on the graph from above. The shadow it casts on the x-axis represents the domain.
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Identify the boundaries: Determine the leftmost and rightmost x-values that the graph occupies. These are the boundaries of your domain.
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Pay attention to endpoints:
- Closed circles (•): Indicate that the endpoint is included in the domain. Use a square bracket "[" or "]" to denote this inclusion.
- Open circles (o): Indicate that the endpoint is not included in the domain. Use a parenthesis "(" or ")" to denote this exclusion.
- Arrows: Indicate that the graph continues indefinitely in that direction. This means the domain extends to positive or negative infinity.
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Consider discontinuities: Look for any breaks, holes, or vertical asymptotes in the graph. These points represent x-values that are not in the domain.
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Express the domain in interval notation: Use interval notation to write down the domain. For example:
[a, b]means all x-values from a to b, including a and b.(a, b)means all x-values from a to b, excluding a and b.[a, ∞)means all x-values from a to positive infinity, including a.(-∞, b)means all x-values from negative infinity to b, excluding b.- If there are multiple intervals, use the union symbol "∪" to combine them. For example:
(a, b) ∪ (c, d)means all x-values in the interval (a,b) or in the interval (c, d).
Example:
Imagine a graph that starts at x = -2 with a closed circle and extends to x = 5 with an open circle. The domain would be [-2, 5).
Key takeaway: The domain is the set of all possible x-values represented on the graph. Visualizing the projection of the graph onto the x-axis is a helpful technique.
